Let I = ex2/2 dx. (a) Show that for any and is equivalent to I = 2. (b) Show that

Let I =

∞

−∞ e−x2/2 dx.

(a) Show that for any μ and σ

is equivalent to I = √
2π.

(b) Show that I = √
2π by writing

and then evaluating the double integral by means of a change of variables to polar coordinates. (That is, let x = r cos θ, y = r sin θ, dx dy = r dr dθ.)

8 1 -(x-1) 120 dx = 1 2 1000